Private Interdependent Valuations: New Bounds for Single-Item Auctions and Matroids

We study auction design within the widely acclaimed model of interdependent values, introduced by Milgrom and Weber 1982. In this model, every bidder i has a private signal sᵢ for the item for sale, and a public valuation function vᵢ (s₁, , sₙ) which maps every vector of private signals (of all bidders) into a real value. A recent line of work established the existence of approximately-optimal mechanisms within this framework, even in the more challenging scenario where each bidder's valuation function vᵢ is also private. This body of work has primarily focused on single-item auctions with two natural classes of valuations: those exhibiting submodularity over signals (SOS) and d-critical valuations. In this work we advance the state of the art on interdependent values with private valuation functions, with respect to both SOS and d-critical valuations. For SOS valuations, we devise a new mechanism that gives an improved approximation bound of 5 for single-item auctions. This mechanism employs a novel variant of an "eating mechanism", leveraging LP-duality to achieve feasibility with reduced welfare loss. For d-critical valuations, we broaden the scope of existing results beyond single-item auctions, introducing a mechanism that gives a (d+1) -approximation for any environment with matroid feasibility constraints on the set of agents that can be simultaneously served. Notably, this approximation bound is tight, even with respect to single-item auctions.